Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,14,17,15,12,13,23)(11,18,22,19,20,24,21,16);(5,6)(7,8)(9,11,13,21,15,20,14,22)(10,16,17,18,12,19,23,24);(2,4)(6,8)(9,12)(10,15)(11,19)(13,23)(14,17)(16,20)(18,22)(21,24);(11,20)(16,19)(18,24)(21,22)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238739, #72238743, #72238755, … |
| maximal normal subgroup of | #72238739, #72238743, #72238755, … |
| maximal quotient of | #72239004 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264886);